Problem Sheet 1. You may assume that both F and F are σ-fields. (a) Show that F F is not a σ-field. (b) Let X : Ω R be defined by 1 if n = 1

Size: px

Start display at page:

Download "Problem Sheet 1. You may assume that both F and F are σ-fields. (a) Show that F F is not a σ-field. (b) Let X : Ω R be defined by 1 if n = 1"

Scarlett Stephens
5 years ago
Views:

1 Problem Sheet 1 1. Let Ω = {1, 2, 3}. Let F = {, {1}, {2, 3}, {1, 2, 3}}, F = {, {2}, {1, 3}, {1, 2, 3}}. You may assume that both F and F are σ-fields. (a) Show that F F is not a σ-field. (b) Let X : Ω R be defined by 1 if n = 1 X(n) = 2 if n = 2 1 if n = 3 Is X measurable with respect to F? Is X measurable with respect to F? 2. Let Ω be any set. Let I be any set and for each i I let F i be a σ-field on Ω. Prove that i I F i is a σ-field on Ω. 3. Let (Ω, F, P) be a probability space and let X : Ω R be a random variable. (a) Show that X 2 is a random variable. (b) Let g : R R be such that g(x) g(y) for all x y. Show that g(x) is a random variable. 4. Let Ω = {0, 1} N, and let us write each ω Ω as a sequence: ω = ω 1 ω 2 ω 3... where ω i {0, 1}. Let F = σ({ω ; ω n = C} ; n N, C {0, 1}). For each n N let X n : Ω R be given by X n (ω) = ω n and define n S n = X i. i=1 (a) Show that the following events are F measurable: { n, X n = 1}, { N, n N, X n = 0}, { sup S m n }. m n 2 Suppose additionally that P : F [0, 1] is a probability measure, under which the X i are independent and identically distributed with P [X 1 = 0] = P [X 1 = 1] = 1 2. (c) Calculate E[S 2 σ(x 1 )] and E[S2 2 σ(x 1 )]. (d) Let n N. Calculate E[X 1 σ(s n )]. 5. Let (Ω, F, P) be a probability space and let X L 1. Let G be a sub-σ-field of F. (a) Prove that E[E[X G]] = E[X] almost surely. (b) Let F 0 = {, Ω}. Prove that there exists c R such that E[X F 0 ] = c almost surely. Hence, show that c = E[X]. 6. Let (Ω, F, P) be a probability space and let X, Y L 1. Let G be a sub-σ-field of F. Suppose that E [X G] = Y and E[X 2 ] = E[Y 2 ]. Prove that X = Y almost surely. 1

2 Problem Sheet 2 1. Let (X n ) n N be an iid sequence of random variables such that P [X 1 = 1] = P [X 1 = 1] = 1 2. Let n S n = X i. Let F n = σ(x i ; i n). (a) Show that F n is a filtration and that S n is a F n martingale. (b) State, with proof, which of the following processes are F n martingales: (i) S 2 n (ii) S 2 n n (iii) S n n Which of the above are submartingales? 2. Let X 0, X 1,... be a sequence of L 1 random variables. Let F n be a filtration and suppose that E[X n+1 F n ] = ax n + bx n 1 for all n N, where a, b > 0 and a + b = 1. Find a value of α R for which S n = αx n + X n 1 is an F n martingale. 3. At time 0, an urn contains 1 black ball and 1 white ball. At each time n = 1, 2, 3,...,, a ball is chosen from the urn and returned to the urn. At the same time, a new ball of the same colour as the chosen ball is added to the urn. Just after time n, there are n + 2 balls in the urn, of which B n + 1 are black, where B n is the number of black balls added into the urn at or before time n. Let M n = B n + 1 n + 2 be the proportion of balls in the urn that are black, at time n. Note that M n [0, 1]. i=1 (a) Show that, relative to a natural filtration that you should specify, M n is a martingale. (b) Calculate the probability that the first k balls drawn are all black and that the next j balls drawn are all white. (c) Show that P [B n = k] = 1 n+1 for all 0 k n, and deduce that lim n P [M n p] = p for all p [0, 1]. (d) Let T be the number of balls drawn until the first black ball appears. Show that T is a stopping 1 time and use the Optional Stopping Theorem to show that E[ T +2 ] = Let S and T be stopping times with respect to the filtration F n. (a) Show that min(s, T ) and max(s, T ) are stopping times. (b) Suppose S T. Is it necessarily true that T S is a stopping time? 5. Suppose that we repeatedly toss a fair coin, writing H for heads and T for tails. What is the expected number of tosses until we have seen the pattern HT HT for the first time? Give an example of a four letter pattern of {H, T } that has the maximal expected number of tosses, of any four letter pattern, until it is seen. 6. Let m N and m 2. At time n = 0, an urn contains 2m balls, of which m are red and m are blue. At each time n = 1,..., 2m we draw a single ball from the urn; we do not replace it. Therefore, at time n the urn contains 2m n balls. Let N n denote the number of red balls remaining in the urn at time n. For n = 0,..., 2m 1 let P n = N n 2m n be the fraction of red balls remaining after time n. Let G n = σ(n i ; i n). (a) Show that P n is a G n martingale. (b) Let T be the first time at which the ball that we draw is red. Note that T < 2m, because the urn initially contains at least 2 red balls. Show that the probability that the (T + 1) st ball is red is

3 Problem Sheet 3 1. You play a game by betting on outcome of i.i.d. random variables X n, n Z +, where 1 P[X n = 1] = p, P[X n = 1] = q = 1 p, 2 < p < 1. Let Z n be your fortune at time n, that is Z n = Z 0 + n j=1 C jx j. The bet C n you place on game n must be in (0, Z n 1 ) (i.e. you cannot borrow money to place bets). Your objective is to maximise the expected interest rate E[log(Z N /Z 0 )], where N (the length of the game) and Z 0 (your initial fortune) are both fixed. Let F n = σ(x 1,..., X n ). Show that if C is a previsible strategy, then log Z n nα is a supermartingale, where and deduce that E log[z n /Z 0 ] Nα. α = p log p + q log q + log 2, Can you find a strategy such that log Z n nα is a martingale? 2. Let F n be a filtration. Suppose T is a stopping time such that for some K 1 and ɛ > 0 we have, for all n 0, almost surely P [T n + K F n ] ɛ. (a) Prove by induction that for all m N, P [T mk] (1 ɛ) m. (b) Hence show that E[T ] <. 3. Let X 1, X 2,... be a sequence of iid random variables with P[X 1 = 1] = p, P[X 1 = 1] = q, where 0 < p = 1 q < 1, and suppose that p q. Let a, b N with 0 < a < b, and let Let F n = σ(x i ; i n). S n = a + X X n, T = inf{n 0 ; S n = 0 or S n = b}. (a) Deduce from the previous question that E[T ] <. (b) Show that ( ) Sn q M n =, N n = S n n(p q) p are both F n martingales. (c) Calculate P[S T = 0], E[S T ] and hence calculate E[T ]. 4. Let X 1, X 2,... be strictly positive iid random variables such that E[X 1 ] = 1 and P[X 1 = 1] < 1. (a) Show that M n = n i=1 X i is a martingale relative to a natural filtration that you should specify. (b) Deduce that there exists a real valued random variable L such that M n L almost surely as n. (c) Show that P[L = 0] = 1. Hint: Argue by contradiction and note that if M n, M n+1 (c ɛ, c+ɛ) then X n+1 ( c ɛ c+ɛ, c+ɛ c ɛ ). (d) Use the Strong Law of Large Numbers to show that there exists c R such that 1 n log M n c almost surely n. Use Jensen s inequality to show that c < Show that a set C of random variables is uniformly integrable if either: (a) There exists a random variable Y such that E[ Y ] < and X Y for all X C. (b) There exists p > 1 and A < such that E[ X p ] A for all X C. 6. Let Z n be a Galton-Watson process with offspring distribution G (which takes value in 0, 1,...), where E[G] = µ > 1 and var[g] = σ 2 <. Set M n = Zn µ n, and use the filtration from lecture notes. Show that M n is a martingale. Find a formula E[M 2 n] in terms of n, µ and σ. Hence, show that sup n N E[M 2 n] < and that M n converges almost surely and in L 1 as n. Deduce that the limit M satisfies P[M > 0] > 0. 3

Martingale Theory for Finance

Martingale Theory for Finance Tusheng Zhang October 27, 2015 1 Introduction 2 Probability spaces and σ-fields 3 Integration with respect to a probability measure. 4 Conditional expectation. 5 Martingales.